Month: August 2017

In Which Bases is 838 a Palindrome?

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838 is a palindrome in base 10. Is it a palindrome in any other bases? Yes, two others.

  • 262 BASE 19 because 2(19²) + 6(19¹) + 2(19º) = 838
  • 141 BASE 27 because 1(27²) + 4(27¹) + 1(27º) = 838

There is only one way 838 can be written as the sum of consecutive numbers:

  • 208 + 209 + 210 + 211 = 838

Print the puzzles or type the solution on this excel file: 10-factors-835-842

  • 838 is a composite number.
  • Prime factorization: 838 = 2 × 419
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 838 has exactly 4 factors.
  • Factors of 838: 1, 2, 419, 838
  • Factor pairs: 838 = 1 × 838 or 2 × 419
  • 838 has no square factors that allow its square root to be simplified. √838 ≈ 28.9482

 

 

837 and Level 3

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837 is repdigit RR in BASE 30 (R is 27 in base 10). All that means is that 27(30¹) + 27(30º) = 837, which naturally follows from the fact that 27(30 + 1) = 837.

837 has four odd factor pairs, so 837 can be written as the difference of two squares four different ways:

  • 837 × 1 = 837 means 419² – 418² = 837
  • 279 × 3 = 837 means 141² – 138² = 837
  • 93 × 9 = 837 means 51² – 42² = 837
  • 31 × 27 = 837 means 29² – 2² = 837

Hmm…837 is only four numbers away from the next perfect square, 841.

Print the puzzles or type the solution on this excel file: 10-factors-835-842

  • 837 is a composite number.
  • Prime factorization: 837 = 3 × 3 × 3 × 31, which can be written 837 = 3³ × 31
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 837 has exactly 8 factors.
  • Factors of 837: 1, 3, 9, 27, 31, 93, 279, 837
  • Factor pairs: 837 = 1 × 837, 3 × 279, 9 × 93, or 27 × 31
  • Taking the factor pair with the largest square number factor, we get √837 = (√9)(√93) = 3√93 ≈ 28.93095

 

 

Finding Ways to Write 836 as the Sum of Consecutive Numbers

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OEIS.org informs us that 836² = 698,896, a palindrome.

Print the puzzles or type the solution on this excel file: 10-factors-835-842

836 can be written as the sum of 11 consecutive numbers and as the sum of 19 consecutive numbers because 11 and 19 are its odd factors (not including 1) that are less than 41. (Remember 861 is the 41st triangular number.) Notice 836’s factor pairs highlighted in red.

  • 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 = 836; that’s 11 consecutive numbers
  • 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 836; that’s 19 consecutive numbers

836 can be written as the sum of 8 consecutive numbers. Why? Because its factor that is the greatest power of 2 is 4, and because 1 is a factor of 836. Note that 2(4)(1) = 8.

  • 101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 = 836

836 can’t be written as the sum of 2(4)(11) = 88 consecutive numbers or 2(4)(19) = 152 consecutive numbers because neither 88 or 152 is less than 41.

  • 836 is a composite number.
  • Prime factorization: 836 = 2 × 2 × 5 × 41, which can be written 836 = 2² × 11 × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 836 has exactly 12 factors.
  • Factors of 836: 1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836
  • Factor pairs: 836 = 1 × 836, 2 × 418, 4 × 209, 11 × 76, 19 × 44, or 22 × 38
  • Taking the factor pair with the largest square number factor, we get √836 = (√4)(√209) = 2√209 ≈ 28.91366

835 and Level 1

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835 is the hypotenuse of a Pythagorean triple, 501-668-835, which is 167 times (3-4-5).

835 can be written as the difference of two squares two different ways:

  • 418² – 417² = 835
  • 86² – 81² = 835

835 can be written as the sum of consecutive numbers three different ways.

  • 417 + 418 = 835; that’s two consecutive numbers.
  • 165 + 166 + 167 + 168 + 169 = 835; that’s five consecutive numbers.
  • 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 835; that’s ten consecutive numbers.

Print the puzzles or type the solution on this excel file: 10-factors-835-842

  • 835 is a composite number.
  • Prime factorization: 835 = 5 × 167
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 835 has exactly 4 factors.
  • Factors of 835: 1, 5, 167, 835
  • Factor pairs: 835 = 1 × 835 or 5 × 167
  • 835 has no square factors that allow its square root to be simplified. √835 ≈ 28.89636655

834 and Level 6

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834 is the sum of consecutive prime numbers two different ways:

  • 127 + 131 + 137 + 139 + 149 + 151 = 834; that’s six consecutive primes
  • 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 834; that’s fourteen consecutive primes

The ONLY Pythagorean triple that contains the number 834 is 834 – 173888 – 173890.

  • 834 is a composite number.
  • Prime factorization: 834 = 2 × 3 × 139
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 834 has exactly 8 factors.
  • Factors of 834: 1, 2, 3, 6, 139, 278, 417, 834
  • Factor pairs: 834 = 1 × 834, 2 × 417, 3 × 278, or 6 × 139
  • 834 has no square factors that allow its square root to be simplified. √834 ≈ 28.879058

There was a solar eclipse in the United States today. People where I lived were able to experience 91.32% obstruction of the sun. I love this interactive map of today’s eclipse and past and future ones as well.

Several people have taken and shared marvelous pictures of the eclipse.

Here are a few tweets I saw about eclipses on twitter:

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833 and Level 5

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

Wrap Your Mind Around This: 7² × 17 = 833 = 28² + 7²

833 is the hypotenuse of a Pythagorean triple:

  • 392-735-833 calculated from 2(28)(7), 28² – 7², 28² + 7²

833 can be written as the sum of consecutive prime numbers two ways. One of those ways starts with one of its prime factors, 17:

  • 833 is the sum of the seventeen prime numbers from 17 to 83
  • 833 is also the sum of the eleven prime numbers from 53 to 101

Since 833 has three factor pairs where both factors are odd, it can be written as the difference of two squares three different ways:

  • 833 × 1 = 833 means 417² – 416² = 833
  • 119 × 7 = 833 means 63² – 56² = 833
  • 49 × 17 = 833 means 33² – 16² = 833

The 41st triangular number will be 861. We must use less than 41 consecutive numbers if we want to express 833 as the sum of consecutive numbers. 833 has 3 odd factors (1, 7, 17) less than 41. Thus 833 can be written as the sum of 7 consecutive numbers and as the sum of 17 consecutive numbers. Notice 833’s factor pairs below highlighted in red.

  • 833 = 116 + 117 + 118 + 119 + 120 + 121 + 122; that’s 7 consecutive numbers
  • 833 = 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57; that’s 17 consecutive numbers

The factor of 833 that is the highest power of 2 is 1 because 2º=1. Each of those odd factors, (1, 7, 17), times 2 × 1 is still less than 40, so 833 can also be written as the sum of 2 consecutive numbers, the sum of 14 consecutive numbers, and the sum of 34 consecutive numbers:

  • 833 = 416 + 417; that’s 2 consecutive numbers
  • 833 = 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66; that’s 14 consecutive numbers
  • 833 = 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41; that’s 34 consecutive numbers

Here is 833’s factoring information:

  • 833 is a composite number.
  • Prime factorization: 833 = 7 × 7 × 17, which can be written 833 = 7² × 17
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 833 has exactly 6 factors.
  • Factors of 833: 1, 7, 17, 49, 119, 833
  • Factor pairs: 833 = 1 × 833, 7 × 119, or 17 × 49
  • Taking the factor pair with the largest square number factor, we get √833 = (√49)(√17) = 7√17 ≈ 28.861739

832 and Level 4

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

832 has many factors, but it can be written as the sum of consecutive numbers only one way:

  • 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 832; that’s thirteen consecutive numbers.

832 can be written as the difference of two squares five different ways because it has five factor pairs in which both numbers are even:

  • 416 × 2 = 832 means 209² – 207² = 832
  • 208 × 4 = 832 means 106² – 102² = 832
  • 104 × 8 = 832 means 56² – 48² = 832
  • 52 × 16 = 832 means 34² – 18² = 832
  • 32 × 26 = 832 means 29² – 3² = 832

832 is also the sum of two squares:

  • 24² + 16² = 832

832 is the hypotenuse of a Pythagorean triple:

  • 320-768-832 calculated from 24² – 16², 2(24)(16), 24² + 16²
  • 320-768-832 is also 64 times (5-12-13)

832 is repdigit QQ in BASE 31 (Q is 26 base 10). That’s because 26(31) + 26(1) = 832, which is the same as saying 26 × 32 = 832.

  • 832 is a composite number.
  • Prime factorization: 832 = 2 × 2 × 2 × 2 × 2 × 2 × 13, which can be written 832 = 2⁶ × 13
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 832 has exactly 14 factors.
  • Factors of 832: 1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832
  • Factor pairs: 832 = 1 × 832, 2 × 416, 4 × 208, 8 × 104, 13 × 64, 16 × 52, or 26 × 32
  • Taking the factor pair with the largest square number factor, we get √832 = (√64)(√13) = 8√13 ≈ 28.8444102

831 and Level 3

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

Before I write a blog post, I look to see how the post number is expressed in different bases. Today I noticed that 831 is 30333 in BASE 4. I was intrigued by all those 3’s because I knew that 831 = 3 × 277. It seems logical that 277 would be 10111 in BASE 4, it turns out that it is! I looked at 831 in all the bases up to BASE 36. Did any others have only multiples of 3 as its digits? Yes, a few did, so I’ve made a chart of 277 and 831 in those five bases to make comparing them easy. I also used only base 10 numbers and not letters of the alphabet to represent the digits in the other bases. As you look at this chart, remember 3 × 277 = 831.

Why are those the ONLY bases for which 3 times the digits of 277 equals the digits for 831? Because in every other base, at least one of the digits times 3 will be greater than or equal to the base and some complicated carrying will have to take place to determine the digits for 831 in that base.

For example, 277 is palindrome 1 11 1 in BASE 12. Obviously 3 times 1 11 1 is 3 33 3. Since 33 is bigger than 12, we somehow end up with non-palindrome 5 9 3 in BASE 12 for 831. This is how that somehow happened: 33÷12 = 2R9. The 9 becomes the middle digit while the 2 is added to the original 3 to make the first digit, 5.

Here’s a little more about the number 831:

Because 277 is one of its factors, 831 is the hypotenuse of a Pythagorean triple: 345-756-831, which is 3 times primitive (115-252-277).

  • 831 is a composite number.
  • Prime factorization: 831 = 3 × 277
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 831 has exactly 4 factors.
  • Factors of 831: 1, 3, 277, 831
  • Factor pairs: 831 = 1 × 831 or 3 × 277
  • 831 has no square factors that allow its square root to be simplified. √831 ≈ 8270706

 

 

830 I Can Divide These Polynomials By (x – 2) Without Even Looking at Them

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Print the puzzles or type the solution on this excel file: 12 factors 829-834

(x – 2) is a factor of an infinite number of polynomials. I am listing only a small, but very special subset of them here. First look for the pattern that allows us to generate a polynomial from a given number in base 2. Then look for another pattern when the polynomial is divided by (x – 2).

Do you see the patterns? I do.

From the first pattern I know there is a similar special polynomial that ends with -830. AND I know from the second pattern what I will get if I divide THAT polynomial by (x – 2). Now get this: Even though I haven’t seen the polynomial yet, I know what the quotient will be! When the polynomial ending in -830 is divided by (x – 2), it will be. . . .

  • x⁸ + 3x⁷ + 6x⁶ + 12x⁵ + 25x⁴ + 51x³ + 103x² + 207x + 415

And guess what, I’m right! I found the quotient without showing any steps or even looking at what I was dividing.

How did I know what that quotient would be without writing down the problem and doing some division first? Well, not only is this polynomial special, but the quotient is special, too!

All I needed to know was that the last term was -830. I then divided 830 repeatedly by 2. Any time my quotient was an odd number, I subtracted one from it before I divided it again by 2. I repeated the process until I reached zero. That is how I got all my coefficients. Even though I could do this problem without showing any work, I made a gif so you and anyone else can quickly see how I did it, but you’ll have to look sideways at it to see it. Showing steps is ALWAYS a good thing.
Find 830 in BASE 2

make science GIFs like this at MakeaGif
As a side benefit, this is another way to find out what 830 is in BASE 2.

Here’s a little more about the number 830:

  • 830 is a composite number.
  • Prime factorization: 830 = 2 × 5 × 83
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 830 has exactly 8 factors.
  • Factors of 830: 1, 2, 5, 10, 83, 166, 415, 830
  • Factor pairs: 830 = 1 × 830, 2 × 415, 5 × 166, or 10 × 83
  • 830 has no square factors that allow its square root to be simplified. √830 ≈ 28.80972

830 is the sum of four consecutive prime numbers:

  • 197 + 199 + 211 + 223 = 830

Because 5 is one of its factors, 830 is the hypotenuse of a Pythagorean triple:

  • 498-664-830; that’s 166 times (3-4-5)

 

829 Is the 24th Centered Triangular Number

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Print the puzzles or type the solution in this excel file: 12 factors 829-834

Whether you write today’s date as 8-15-17 or 15-8-17, today is (Primitive) Pythagorean Triple Day. 829 is also in a couple of primitive Pythagorean triples:

  • 540 – 629 – 829, calculated from 2(27)(10), 27² – 10², 27² + 10²
  • 829 – 343620 – 343621, calculated from (829)(1), (829² – 1²)/2, (829² + 1²)/2

829 is the last prime number in the fourth prime decade, (821, 823, 827, 829). Later this year we can celebrate the FIRST prime decade on 11-13-17 at 1900 hours.

Sara Van Der Werf wrote a post that includes a poster of these and other mathematical holidays that can be celebrated in the 2017-18 school year. She will also keep us posted on ways to celebrate these holidays as each one approaches. The poster is a word document that can be edited if need be.

829 is also the sum of three consecutive prime numbers: 271 + 277 + 281 = 829

829 is the 24th centered triangular number. I challenged myself to make a graphic that would show the significance of this fun number fact.

Now you can see why it is called a CENTERED triangular number. If you count these concentric triangles including the tiny, barely visible one in the very center, you will see that there are 24 of them. Can you also see that 829 is equal to the sum of the 22nd, 23rd, and 24th triangular numbers?

What else is true because 829 is the 24th centered triangular number?

  •  (22×23 + 23×24 + 24×25)/2 = 829.
  • 3(23×24)/2 + 1 = 829

Here is 829’s factoring information:

  • 829 is a prime number.
  • Prime factorization: 829 is prime.
  • The exponent of prime number 829 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 829 has exactly 2 factors.
  • Factors of 829: 1, 829
  • Factor pairs: 829 = 1 × 829
  • 829 has no square factors that allow its square root to be simplified. √829 ≈ 28.79236

How do we know that 829 is a prime number? If 829 were not a prime number, then it would be divisible by at least one prime number less than or equal to √829 ≈ 28.8. Since 829 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 829 is a prime number.

Here’s another way we know that 829 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 10² = 829 with 27 and 10 having no common prime factors, 829 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √829 ≈ 28.8. Since 829 is not divisible by 5, 13, or 17, we know that 829 is a prime number.